Recess sink into wooden worktop

I've got a solid wood worktop, and a stainless steel sink with a lip that sits on top of the worktop. This is fine except water often pools between the sink and the wall which I'd like to solve. Often so much water pools there from people carelessly washing their hands that it flows out onto the main worktop area! I can only prompt people so much to wipe up this water!

The lip of the sink only protrudes about 2 mm above the surface. I was thinking of routing the surface such that the lip of...

Recess sink into wooden worktop